# American Institute of Mathematical Sciences

November  2017, 37(11): 5883-5912. doi: 10.3934/dcds.2017256

## Visco-Energetic solutions to one-dimensional rate-independent problems

 Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy

Received  October 2016 Revised  June 2017 Published  July 2017

Visco-Energetic solutions of rate-independent systems (recently introduced in [17]) are obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation is reinforced by a viscous correction $δ$, typically a quadratic perturbation of the dissipation distance. Like Energetic and Balanced Viscosity solutions, they provide a variational characterization of rate-independent evolutions, with an accurate description of their jump behaviour.

In the present paper we study Visco-Energetic solutions in the scalar-valued case and we obtain a full characterization for a broad class of energy functionals. In particular, we prove that they exhibit a sort of intermediate behaviour between Energetic and Balanced Viscosity solutions, which can be finely tuned according to the choice of the viscous correction $δ$.

Citation: Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000.  Google Scholar [2] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799.  doi: 10.1142/S0218202502002331.  Google Scholar [3] M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Analysis, 13 (2006), 151-167.   Google Scholar [4] I. S. Gál, On the fundamental theorems of the calculus, Trans. Amer. Math. Soc., 86 (1957), 309-320.  doi: 10.1090/S0002-9947-1957-0093562-7.  Google Scholar [5] D. Knees and A. Schröder, Computation aspect of quasi-static crack propagation, DCDS-S, 6 (2013), 63-99.  doi: 10.3934/dcdss.2013.6.63.  Google Scholar [6] K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl.(4), 40 (1955), 61-67.  doi: 10.1007/BF02416522.  Google Scholar [7] D. Leguillon, Strength or toughness? A criterion for crack onset at a notch, European J. of Mechanics A/Solids, 21 (2002), 61-72.  doi: 10.1016/S0997-7538(01)01184-6.  Google Scholar [8] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.  Google Scholar [9] A. Mielke, Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDEs and Applications, Lect. Notes Math, Springer, 2028 (2011), 87-170.  doi: 10.1007/978-3-642-21861-3_3.  Google Scholar [10] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems A, 25 (2009), 585-615.  doi: 10.3934/dcds.2009.25.585.  Google Scholar [11] Alexander Mielke, Riccarda Rossi and Giuseppe Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80.  doi: 10.1051/cocv/2010054.  Google Scholar [12] Alexander Mielke, Riccarda Rossi and Giuseppe Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc., 18 (2016), 2107-2165.  doi: 10.4171/JEMS/639.  Google Scholar [13] A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar [14] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.  doi: 10.1007/s00030-003-1052-7.  Google Scholar [15] A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.  doi: 10.1007/s002050200194.  Google Scholar [16] L. Minotti, Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016. Google Scholar [17] L. Minotti and G. Savaré, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems, arXiv: 1606.03359, (2016), 1-60. Google Scholar [18] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925.  doi: 10.1142/S0218202508003236.  Google Scholar [19] R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 97-169.   Google Scholar [20] R. Rossi and G. Savaré, A characterization of energetic and BV solutions to one-dimensional rate-independent systems, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 167-191.   Google Scholar [21] T. Roubíček, C. C. Panagiotopoulos and V. Mantic, Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013), 823-840.  doi: 10.1002/zamm.201200239.  Google Scholar

show all references

##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000.  Google Scholar [2] G. Dal Maso and R. Toader, A model for the quasi-static growth of brittle fractures based on local minimization, Math. Models Methods Appl. Sci., 12 (2002), 1773-1799.  doi: 10.1142/S0218202502002331.  Google Scholar [3] M. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity, J. Convex Analysis, 13 (2006), 151-167.   Google Scholar [4] I. S. Gál, On the fundamental theorems of the calculus, Trans. Amer. Math. Soc., 86 (1957), 309-320.  doi: 10.1090/S0002-9947-1957-0093562-7.  Google Scholar [5] D. Knees and A. Schröder, Computation aspect of quasi-static crack propagation, DCDS-S, 6 (2013), 63-99.  doi: 10.3934/dcdss.2013.6.63.  Google Scholar [6] K. Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl.(4), 40 (1955), 61-67.  doi: 10.1007/BF02416522.  Google Scholar [7] D. Leguillon, Strength or toughness? A criterion for crack onset at a notch, European J. of Mechanics A/Solids, 21 (2002), 61-72.  doi: 10.1016/S0997-7538(01)01184-6.  Google Scholar [8] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  doi: 10.1007/s00526-004-0267-8.  Google Scholar [9] A. Mielke, Differential, energetic and metric formulations for rate-independent processes, Nonlinear PDEs and Applications, Lect. Notes Math, Springer, 2028 (2011), 87-170.  doi: 10.1007/978-3-642-21861-3_3.  Google Scholar [10] A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete and Continuous Dynamical Systems A, 25 (2009), 585-615.  doi: 10.3934/dcds.2009.25.585.  Google Scholar [11] Alexander Mielke, Riccarda Rossi and Giuseppe Savaré, BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var., 18 (2012), 36-80.  doi: 10.1051/cocv/2010054.  Google Scholar [12] Alexander Mielke, Riccarda Rossi and Giuseppe Savaré, Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems, J. Eur. Math. Soc., 18 (2016), 2107-2165.  doi: 10.4171/JEMS/639.  Google Scholar [13] A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar [14] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.  doi: 10.1007/s00030-003-1052-7.  Google Scholar [15] A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177.  doi: 10.1007/s002050200194.  Google Scholar [16] L. Minotti, Visco-Energetic Solutions to Rate-Independent Evolution Problems, PhD thesis, Pavia, 2016. Google Scholar [17] L. Minotti and G. Savaré, Viscous corrections of the time incremental minimization scheme and visco-energetic solutions to rate-independent evolution problems, arXiv: 1606.03359, (2016), 1-60. Google Scholar [18] M. Negri and C. Ortner, Quasi-static crack propagation by Griffith's criterion, Math. Models Methods Appl. Sci., 18 (2008), 1895-1925.  doi: 10.1142/S0218202508003236.  Google Scholar [19] R. Rossi, A. Mielke and G. Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 97-169.   Google Scholar [20] R. Rossi and G. Savaré, A characterization of energetic and BV solutions to one-dimensional rate-independent systems, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 167-191.   Google Scholar [21] T. Roubíček, C. C. Panagiotopoulos and V. Mantic, Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity, Zeitschrift angew. Math. Mech., 93 (2013), 823-840.  doi: 10.1002/zamm.201200239.  Google Scholar
The double-well potential $W$ with its convex envelope in bold (left picture) and an energetic solution $u$ in the case of a strictly increasing load $\ell$ (right picture).
BV solution for a double-well energy $W$ with an increasing load $\ell$. The blue line denotes the path described by the optimal transition $\vartheta$ solving (4).
Visco-Energetic solutions for a double-well energy $W$ with an increasing load $\ell$. When $\mu>-\min W''$ (left picture) the solution jumps when it reach the maximum of $W'$ and the transition is the ''double chain'' obtained by solving the Incremental Minimization Scheme with frozen time $t$. When $\mu$ is small (right picture) the optimal transition $\vartheta$ makes a first jump connecting $\mathit{u}_{\tiny \mathsf L}(t)$ with $u_+$ according to the modified Maxwell rule (9): $\mathit{u}_{\tiny \mathsf L}(t)$ and $u_+$ corresponds to the intersection of $W'$ with the red line, whose slope is $-\mu$.
The one-sided slopes and the stability region when $W$ is a double-well potential. For some suitable choices of $\delta$, $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ is intermediate between $W'_{\mathsf i\mathsf r}$ and $W'$.
$\mathsf D$-Maxwell's rule for a double well potential: when $\delta=\frac{\mu}{2}|\cdot|^2$, the ''last point'' where $W'$ and $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ coincide is such that the total area between the graph $W'$ and the line whose slope is $-\mu$ is zero.
Visco-Energetic solution of a double-well potential energy with an oscillating external loading and a quadratic viscous-correction $\delta(u,v)$, turned by a parameter $\mu>-\min W''$.
Visco-Energetic solutions of a nonconvex energy and an increasing loading. The optimal transition is a combination of sliding and viscous parts.
 [1] Riccarda Rossi, Giuseppe Savaré. A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 167-191. doi: 10.3934/dcdss.2013.6.167 [2] Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020332 [3] Alexander Mielke, Riccarda Rossi, Giuseppe Savaré. Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 585-615. doi: 10.3934/dcds.2009.25.585 [4] Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020304 [5] Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 [6] Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i [7] T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215 [8] Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257 [9] Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649 [10] Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070 [11] Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145 [12] Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i [13] Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo, Herme Soto. Existence of solutions for a class of abstract neutral differential equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2455-2482. doi: 10.3934/dcds.2017106 [14] Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks & Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257 [15] Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591 [16] Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411 [17] Robert Hesse, Alexandra Neamţu. Global solutions and random dynamical systems for rough evolution equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2723-2748. doi: 10.3934/dcdsb.2020029 [18] Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291 [19] Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455 [20] Alessandro Giacomini. On the energetic formulation of the Gurtin and Anand model in strain gradient plasticity. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 527-552. doi: 10.3934/dcdsb.2012.17.527

2018 Impact Factor: 1.143

## Metrics

• PDF downloads (33)
• HTML views (73)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]